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Diffusive Expansion of the Boltzmann equation for the flow past an obstacle

Yan Guo, Junhwa Jung

Abstract

The exterior domain problem is essential in fluid and kinetic equations. In this paper, we establish the validity of the diffusive expansion for the Boltzmann equations to the Navier-Stokes-Fourier system up to the critical time in an exterior domain with non-zero passing flow. We apply the $L^3-L^6$ framework to the unbounded domain in this paper.

Diffusive Expansion of the Boltzmann equation for the flow past an obstacle

Abstract

The exterior domain problem is essential in fluid and kinetic equations. In this paper, we establish the validity of the diffusive expansion for the Boltzmann equations to the Navier-Stokes-Fourier system up to the critical time in an exterior domain with non-zero passing flow. We apply the framework to the unbounded domain in this paper.

Paper Structure

This paper contains 13 sections, 17 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Notation Definition
  3. Boundary condition
  4. Main Result
  5. Historical background
  6. Linear Estimate
  7. Energy estimate
  8. $L^p$ estimate
  9. Nonlinear estimate
  10. Estimate of Navier-Stokes equation
  11. Estimate of fluid term
  12. Estimate of nonlinear term
  13. Proof of the main theorem

Key Result

Theorem 2

Let $\Omega$ be a $C^\infty$ bounded open set in $\mathbb{R}^3$, and let $\Omega^c = \mathbb{R}^3 \backslash \bar{\Omega}$. For any $0< M$, consider the following boundary value problem: Suppose the initial datum takes the form $F_0 = \mu_\mathfrak{u} + \varepsilon \sqrt{\mu_\mathfrak{u} } f_0 \ge 0$ such that and satisfies where $\omega(v) = e^{\beta \left\lvert v\right\rvert^2}$ with $0 < \

Theorems & Definitions (41)

  • Definition 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Remark 8
  • ...and 31 more